Higher Rank $\hat{Z}$ and $F_K$

We study $q$-series-valued invariants of $3$-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov–Pei–Putrov–Vafa [arXiv:1701.06567] and Gukov–Manolescu [arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full mathematical definition for these “invariants” is lacking yet, we define $\hat{Z}^G$ for negative definite plumbed $3$-manifolds and $F_K^G$ for torus knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu, there is a surgery formula relating $F_K^G$ to $\hat{Z}^G$ of a Dehn surgery on the knot $K$. Furthermore, specializing to symmetric representations, $F_K^G$ satisfies a recurrence relation given by the quantum $A$-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these $3$-manifold invariants.